The prob
lem of finding a comp
lete
linear description of the stab
le set po
lytope of c
law-free graphs has been a major topic in combinatoria
l optimization in the
last decades and it is sti
ll not comp
lete
ly so
lved. Whi
le it is known that this
linear description contains facet defining inequa
lities with arbitrari
ly many and arbitrari
ly high coefficients, the set of c
law-free graphs whose stab
le set po
lytope is described on
ly by inequa
lities with
lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003490&_mathId=si3.gif&_user=111111111&_pii=S0012365X15003490&_rdoc=1&_issn=0012365X&md5=83f1084b0c88c6faba6beb6bf04794f1" title="Click to view the MathML source">{0,1,2}lass="mathContainer hidden">lass="mathCode">-va
lued coefficients is considerab
ly
large. In fact Ga
lluccio et a
l. (2014) proved that this set contains a
lmost a
ll c
law-free graphs with stabi
lity number greater than three p
lus some of the bui
lding b
locks with stabi
lity number sma
ller than or equa
l to three, identified by Chudnovsky and
Seymour in their Structure Theorem.
Here we show that another important class of claw-free graphs with stability number three belongs to this set: the class of icosahedral graphs, named lsi4" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003490&_mathId=si4.gif&_user=111111111&_pii=S0012365X15003490&_rdoc=1&_issn=0012365X&md5=3fbec9a3ca6e828c8eace8a1bf3c4b77" title="Click to view the MathML source">S1lass="mathContainer hidden">lass="mathCode"> by Chudnovsky and Seymour (2008). In particular, we prove that the stable set polytope of icosahedral graphs is described by: rank, lifted 5-wheel and lifted wedge inequalities, and all these linear inequalities have coefficients in lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15003490&_mathId=si3.gif&_user=111111111&_pii=S0012365X15003490&_rdoc=1&_issn=0012365X&md5=83f1084b0c88c6faba6beb6bf04794f1" title="Click to view the MathML source">{0,1,2}lass="mathContainer hidden">lass="mathCode">.